We consider the problem of state estimation from a few linear measurements, where the state to recover is an element of the manifold $\mathcal{M}$ of solutions of a parameter-dependent equation. The state is estimated using prior knowledge on $\mathcal{M}$ coming from model order reduction. Variational approaches based on linear approximation of $\mathcal{M}$, such as PBDW, yields a recovery error limited by the Kolmogorov width of $\mathcal{M}$. To overcome this issue, piecewise-affine approximations of $\mathcal{M}$ have also been considered, that consist in using a library of linear spaces among which one is selected by minimizing some distance to $\mathcal{M}$. In this paper, we propose a state estimation method relying on dictionary-based model reduction, where a space is selected from a library generated by a dictionary of snapshots, using a distance to the manifold. The selection is performed among a set of candidate spaces obtained from a set of $\ell_1$-regularized least-squares problems. Then, in the framework of parameter-dependent operator equations (or PDEs) with affine parametrizations, we provide an efficient offline-online decomposition based on randomized linear algebra, that ensures efficient and stable computations while preserving theoretical guarantees.

翻译：我们考虑从少量线性测量值进行状态估计的问题，其中待恢复的状态是参数依赖方程解流形 $\mathcal{M}$ 上的一个元素。利用模型降阶方法获取的关于 $\mathcal{M}$ 的先验知识进行状态估计。基于线性近似 $\mathcal{M}$ 的变分方法（如PBDW）的恢复误差受限于 $\mathcal{M}$ 的Kolmogorov宽度。为克服此局限，研究者还提出了分段仿射近似 $\mathcal{M}$ 的方法，即使用一个线性空间库，通过最小化到 $\mathcal{M}$ 的某种距离来选取其中一个空间。本文提出一种基于字典模型降阶的状态估计方法，该方法通过快照字典生成线性空间库，并利用到流形的距离从中选取空间。该选取过程在一组由 $\ell_1$ 正则化最小二乘问题获得的候选空间中进行。随后，在具有仿射参数化的参数依赖算子方程（或偏微分方程）框架下，我们提出一种基于随机线性代数的高效离线-在线分解方法，该方法在确保计算效率与稳定性的同时，保持了理论保证。